四阶离散边值问题正解的存在性

Existence of positive solutions for a fourth order discrete boundary value problem

讨论四阶离散边值问题{Δ4 u(t-2)=f(t,u(t)),t∈T2,u(1)=u(T+1)=Δ2 u(0)=Δ2 u(T)=0正解的存在性,其中f:T2×[0,∞)→(-∞,+∞)是连续且下方有界的,T是大于或等于5的正整数,T2={2,3,…,T}.通过线性和算子谱的性质获得正解的先验估计,在此基础上,借助Krasnoselskii-Zabreiko不动点定理给出了四阶离散边值问题正解的存在性结果.

The paper is concerned with the existence of positive solutions for the fourth order discrete boundary value problemΔ4 u（t -2） = f （t ,u（t）） , t ∈ T2 , u（1） = u（T ＋ 1） = Δ2 u（0） = Δ2 u（T） = 0 , where f ：T2 ＆#215; [0 ,∞） （- ∞ ,＋ ∞） is continuous and bounded below , T is an integer with T≥5 and T2={2 ,3 ,… ,T} . By use of Krasnoselskii-Zabreiko fixed point theorem and priori estimates of positive solution derived by spectral properties of associated linear summation operators , the existence results of positive solution for the four order discrete boundary value problem is given .